00000243.htm

226 IX ANALYTIC FUNCTIONS

Indeed, suppose \z - 0| < q • d(a, y(I)) with 0 < tf < 1; then, for any
x e y(I)> we have

1 1 « (z-a)

— ,Z

-a\ n^o(x-ar¹⁵

with

(z-a)»

\n+l

'«*'

if 5 = rf(a, v(I)). If |
tel,

M in y(I) and \y'(t)\ ^ m in I, we have, for any

<____^w

^ s ^

hence the series of general term

- a)"

\n+l

is normally convergent in I. It follows from (8.7.9) that the series (c_n(z - a)ⁿ)
is convergent in the ball \z - a| < q * 8 and has sum/0) in that ball.

(9.9.3) Under the assumptions of (9.9.1), we have, for every xeA~y(T),
and every integer k > 0,

^fc!

This follows at once from Cauchy's formula, the uniqueness of the coeffi-
cients of a power series with given sum (9.1.6), the relations (9.3.5) between
these coefficients and derivatives, and finally (9.9.2).

(9.9.4) Let AaC^p be an open set, fa continuous mapping of A into a complex
Banach space E, such that for l^k^p, and an arbitrary point (a_k)&C^p,
the mapping z_k -~*f(a_l9..., a_k^_ly z_k9 a_k+i9..., a_p) is analytic in the open set
A(#_1?..., flfc-i, tffc+i,..., a_p) c C if that set is not empty (notation of
(3.20.12)). Then f is analytic in A. More precisely, let a = (a_k) be a point of
A, P a closedpolydisk of center a and radii r_k(l ^k^p) contained in A; for
each k_y let y_k be the circuit t-+a_k + r_k e^h in C (0 < t < 2n)₉ and let

_dx¹

f(x₁,...,x_p)dx_f



226 IX ANALYTIC FUNCTIONS Indeed, suppose \z - 0\| < q • d(a, y(I)) with 0 < tf < 1; then, for any x e y(I)> we have 1 1 « (z-a)

— ,Z		-a\ n^o(x-ar¹⁵

with

(z-a)»

\n+l			'«*'

if 5 = rf(a, v(I)). If \| tel,	M in y(I) and \y'(t)\ ^ m in I, we have, for any

Mm <____^w ^ s ^

hence the series of general term

- a)"

\n+l

is normally convergent in I. It follows from (8.7.9) that the series (c_n(z - a)ⁿ) is convergent in the ball \z - a\| < q 8* and has sum/0) in that ball.

(9.9.3) Under the assumptions of (9.9.1), we have, for every xeA~y(T), and every integer k > 0, ^fc!

This follows at once from Cauchy's formula, the uniqueness of the coeffi- cients of a power series with given sum (9.1.6), the relations (9.3.5) between these coefficients and derivatives, and finally (9.9.2).

(9.9.4) Let AaC^p be an open set, fa continuous mapping of A into a complex Banach space E, such that for l^k^p, and an arbitrary point (a_k)&C^p, the mapping z_k -~f(a_l9..., a_k^_ly z_k9 a_k+i9..., a_p) is analytic in the open set* A(#_1?..., flfc-i, tffc+i,..., a_p) c C if that set is not empty (notation of (3.20.12)). Then f is analytic in A. More precisely, let a = (a_k) be a point of A, P a closedpolydisk of center a and radii r_k(l ^k^p) contained in A; for each k_y let y_k be the circuit t-+a_k + r_k e^h in C (0 < t < 2n)₉ and let

_dx¹			f(x₁,...,x_p)dx_f